A tour boat crosses a wide river. Which velocity is the tour boat's velocity over the riverbed?
A boat aims straight at the far bank at 4 m/s, and the 3 m/s current carries it downstream as it goes. The boat ends up moving toward the far bank and downstream at once. How fast, and at what angle?
The last lesson gave the rule: the boat's velocity over the riverbed is its velocity through the water plus the current's velocity. Here the boat's heading is perpendicular to the current. So the two velocities form the legs of a right triangle, and the resultant is its hypotenuse.
The legs are perpendicular, so the Pythagorean theorem gives the resultant's magnitude:
resultant speed = √(4² + 3²) = √25 = 5 m/s
The tangent gives the angle between the boat's heading and its actual track:
tan θ = 3 ÷ 4, so θ = tan⁻¹(3 ÷ 4) ≈ 37°
The boat travels at 5 m/s, angled 37° downstream of its heading.
A launch aims straight at the far bank at 6 m/s through the water, in a 2.5 m/s current. Calculate the magnitude of its resultant velocity.
Write down the values in the question:
velocity through the water = 6 m/s, perpendicular to the current
current = 2.5 m/s
resultant speed = ?
Write down the equation:
resultant speed = √(heading speed² + current speed²)
Substitute in the values:
resultant speed = √(6² + 2.5²)
Simplify:
resultant speed = √42.25 = 6.5 m/s
The same launch: 6 m/s through the water, 2.5 m/s current. Calculate the angle between its heading and its track over the riverbed.
Write down the values in the question:
velocity through the water = 6 m/s, perpendicular to the current
current = 2.5 m/s
θ = ?
Write down the equation:
tan θ = current speed ÷ heading speed
Substitute in the values:
tan θ = 2.5 ÷ 6
Simplify:
θ = tan⁻¹(2.5 ÷ 6) = 22.6° downstream of the heading
The same routine serves a plane in a crosswind: heading velocity and wind velocity as the legs, ground velocity as the hypotenuse.
A kayak aims perpendicular to a 2.0 m/s current, paddling 2.1 m/s through the water. Enter the kayak's resultant speed in m/s, to one decimal place.
A kayak aims perpendicular to a 2.0 m/s current, paddling 2.1 m/s through the water. Enter the angle between its heading and its track, in degrees to one decimal place.
A plane heads due north at 60 m/s through the air while an 11 m/s wind blows due east. Enter the plane's ground speed in m/s, to one decimal place.
A launch aims perpendicular to a 1.2 m/s current, at 3.5 m/s through the water. Enter its resultant speed in m/s, to one decimal place.
A rowboat aims perpendicular to a 0.7 m/s current, at 2.4 m/s through the water. Enter the angle between its heading and its track over the riverbed, in degrees to one decimal place.
A drone flies at 16 m/s through the air, headed perpendicular to a 12 m/s crosswind. Enter the drone's ground speed in m/s, to one decimal place.